Ultra-wide bandwidth technology (UWB) is currently being investigated as a promising solution for high capacity wireless multiple access systems. A time-hopping sequence is applied in UWB systems to eliminate catastrophic collisions in multiple access deployments. Studies of multiple access system performance for time-hopping systems have been conducted in which a conventional single-user matched filter (correlation receiver) was used to detect a desired user signal. It has been shown that multiple access interference significantly degrades the bit error rate (BER). In some studies, the bit error rate (BER) was estimated by using a Gaussian approximation in which a central limit theorem (CLT) was employed to approximate the sum of multiple access interference (MAI) as an additive Gaussian noise (AGN) process. If a signal is corrupted by AGN, the matched filter is an optimum receiver in the sense that it maximizes an output signal-to-noise ratio (SNR). In the absence of intersymbol interference, it is also the minimum probability of error receiver. However, the MAI in time-hopping ultra-wide bandwidth technology (TH-UWB) systems is not Gaussian-distributed interference. The Gaussian approximation significantly underestimates the BER of practical TH-UWB systems for medium and large SNR values, where the power of the MAI is large. In other words, multiple access interference in TH-UWB systems cannot be reliably modeled as AGN. Therefore, the conventional single-user matched filter or correlation receiver is not necessarily an optimal single-user receiver for UWB. Furthermore, in applications, where it is desired to achieve maximum user capacity, the performance of the system will be limited by MAI and the Gaussian noise may be negligible.
Time-Hopping UWB System Models
In the detailed examples presented below, a time-hopping binary phase shift keying (TH-BPSK) UWB system is considered, but the analysis can also be used for time-hopping pulse position modulation (TH-PPM) systems. A typical TH-BPSK UWB signal has the form
                                          s                          (              k              )                                ⁡                      (            t            )                          ⁢                                            E              b                                      N              s                                      ⁢                              ∑                          j              =                              -                ∞                                      ∞                    ⁢                                    d                              ⌊                                  j                  /                                      N                    s                                                  ⌋                                            (                k                )                                      ⁢                          p              ⁡                              (                                  t                  -                                      jT                    f                                    -                                                            c                      j                                              (                        k                        )                                                              ⁢                                          T                      c                                                                      )                                                                        (        1        )            where t is time, s(k)(t) is the kth user's signal conveying the jth data bit, and p(t) is the signal pulse with pulse width Tp, normalized so that ∫−∞+∞p2(t)=1. The structure of this TH-BPSK model is described as follows:    Eb is the bit energy common to all signals;    Ns is the number of pulses required to transmit a single information bit, also known as a repetition code length;    Tf is the time duration of a frame, and thus, the bit duration Tb=NsTf;    Tc is the hop width satisfying NhTc≦Tf;    {cj(k)} represents the TH code for the kth source; it is pseudorandom with each element taking an integer value in the range 0≦cj(k)<Nh, where Nh is the number of hops;    dj(k) represents the jth binary data bit transmitted by the kth source, taking values from {1,−1} with equal probability.
Assuming Nu users are transmitting asynchronously and the MAI dominates the ambient noise, the received signal is
                              r          ⁡                      (            t            )                          =                              ∑                          k              =              1                                      N              u                                ⁢                                    A              k                        ⁢                                          s                                  (                  k                  )                                            ⁡                              (                                  t                  -                                      τ                    k                                                  )                                                                        (        2        )            where {Ak}k=1Nu represent the channel gains for all transmitted signals, and {τk}k=1Nu represent time shifts which account for user asynchronisms. Without loss of generality, it is assumed that τ1=0. Following a widely-adopted assumption on τk, it is further assumed that {τk}k=2Nu are uniformly distributed on a bit duration (0,Tb], in which Tb defines the length of the bit duration.Conventional Receiver Structures
A conventional single-user matched filter or correlation receiver can be used to coherently demodulate the desired user signal in an asynchronous system. For example, s(1)(t) is assumed to be the reference signal and d0(1)(t) is assumed to be the transmitted symbol. Without loss of generality, cj(1)=0 is set for all j. Assuming perfect synchronization with the reference signal, the decision statistic of the conventional single-user correlation receiver is obtained as
                    r        =                                            ∑                              m                =                0                                                              N                  s                                -                1                                      ⁢                                          ∫                                  mT                  f                                                                      (                                          m                      +                      1                                        )                                    ⁢                                      T                    f                                                              ⁢                                                r                  ⁡                                      (                    t                    )                                                  ⁢                                  p                  ⁡                                      (                                          t                      -                                              τ                        1                                            -                                              mT                        f                                                              )                                                  ⁢                                                                  ⁢                                  ⅆ                  t                                                              =                      S            +            I                                              (        3        )            where S=A1√{square root over (EbNs)}d0(1) depends on one user of a set of Nu users target signal bit d0(1), and I is the total MAI from the Nu−1 remaining active users in the time-hopping binary phase shift keying ultra-wide bandwidth technology (TH-BPSK UWB) system, given by
                              I          =                                                                      E                  b                                                  N                  s                                                      ⁢                                          ∑                                  k                  =                  2                                                  N                  u                                            ⁢                                                A                  k                                ⁢                                  I                                      (                    k                    )                                                                                      ⁢                                  ⁢        where                            (        4        )                                          I                      (            k            )                          =                              ∑                          m              =              0                                                      N                s                            -              1                                ⁢                                    ∫                              mT                f                                                              (                                      m                    +                    1                                    )                                ⁢                                  T                  f                                                      ⁢                                                            s                                      (                    k                    )                                                  ⁡                                  (                                      t                    -                                          τ                      k                                                        )                                            ⁢                                                p                  ⁡                                      (                                          t                      -                                              τ                        1                                            -                                              mT                        f                                                              )                                                  ⁢                                                                  .                                                                        (        5        )            The difference of time shifts for user asynchronism is modeled asτk−τ1=mkTf+αk, −Tf/2≦αk<Tf/2  (6)where mk is the value of the time difference τk−τ1 rounded to the nearest frame time, and αk is uniformly distributed on [−Tf/2, Tf/2). Based on the assumption
                                          N            h                    ⁢                      T            c                          <                                            T              f                        2                    -                      2            ⁢                                                  ⁢                          T              p                                                          (        7        )            Eq. (5) can be re-written in the form
                              I                      (            k            )                          =                              ∑                          m              =              0                                                      N                s                            -              1                                ⁢                                          ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                          d                                  ⌊                                                            (                                              m                        +                                                  m                          k                                                                    )                                        /                                          N                      s                                                        ⌋                                                  (                  k                  )                                            ⁢                              p                ⁡                                  (                                      x                    -                                          α                      k                                        -                                                                  c                        m                                                  (                          k                          )                                                                    ⁢                                              T                        c                                                                              )                                            ⁢                              p                ⁡                                  (                  x                  )                                            ⁢                                                          ⁢                              ⅆ                x                                                                        (        8        )            and the MAI, I, can be expressed as
                    I        =                                                            E                b                                            N                s                                              ⁢                                    ∑                              k                =                2                                            N                u                                      ⁢                                                  ⁢                                          A                k                            ⁢                                                ∑                                      m                    =                    0                                                                              N                      s                                        -                    1                                                  ⁢                                                                  ⁢                                                      d                                          ⌊                                                                        (                                                      m                            +                                                          m                              k                                                                                )                                                /                                                  N                          s                                                                    ⌋                                                              (                      k                      )                                                        ⁢                                                            R                      ⁡                                              (                                                                              α                            k                                                    +                                                                                    c                              m                                                              (                                k                                )                                                                                      ⁢                                                          T                              c                                                                                                      )                                                              .                                                                                                          (        9        )            Then, the desired data symbol can be detected based on the output of the conventional single-user correlation receiver.
It is seen from Eq. (9) that the decision statistic is obtained with a summation of integrals over the number of frames required to transmit one information bit, Ns. Each integration is a partial correlation for the corresponding frame. The decision statistic r can be rewritten as
                    r        =                                            ∑                              m                =                0                                                              N                  s                                -                1                                      ⁢                          r              m                                =                                    ∑                              m                =                0                                                              N                  s                                -                1                                      ⁢                          (                                                S                  m                                +                                  I                  m                                            )                                                          (        10        )            where Sm is the desired signal component on the mth frame, given by
            S      m        =                  A        1            ⁢                                    E            b                                N            s                              ⁢              d        0                  (          1          )                      ,and Im is the MAI on the mth frame, given by
                              I          m                =                                                            E                b                                            N                s                                              ⁢                                    ∑                              k                =                2                                            N                u                                      ⁢                                                  ⁢                                          A                k                            ⁢                              d                                  ⌊                                                            (                                              m                        +                                                  m                          k                                                                    )                                        /                                          N                      s                                                        ⌋                                                  (                  k                  )                                            ⁢                                                R                  ⁡                                      (                                                                  α                        k                                            +                                                                        c                          m                                                      (                            k                            )                                                                          ⁢                                                  T                          c                                                                                      )                                                  .                                                                        (        11        )            The output of the conventional correlation receiver, r, is the sum of the partial correlations on each frame.